A potential function is given by U(x) = 18 x^2. What will be the acceleration (in meters/second ^2) of a 8.00 kg mass, when it is at the position x = 0.500 m?
That U(x) function results when the spring constant is 36. I am assuming that U is in joules.
At position x = 0.50, the force on the mass is 18 J.
Divide that by 8.0 kg for the acceleration.
To find the acceleration of a mass at a given position using a potential function, we need to take the second derivative of the potential function with respect to position.
Given the potential function U(x) = 18x^2, let's first find the force acting on the mass at the position x.
The force can be obtained by taking the derivative of the potential function with respect to position:
F(x) = -dU(x)/dx
Differentiating U(x) = 18x^2 with respect to x, we get:
F(x) = -d/dx (18x^2)
= -36x
Now, we need to apply Newton's second law of motion, which states that force equals mass times acceleration:
F(x) = m * a
where F(x) is the force, m is the mass, and a is the acceleration.
Substituting the value of the force we found earlier, F(x) = -36x, and the given mass m = 8.00 kg, we can solve for the acceleration a.
-36x = m * a
-36(0.500) = 8.00 * a
-18 = 8.00 * a
a = -18 / 8.00
a ≈ -2.25 m/s^2
Therefore, the acceleration of the 8.00 kg mass at x = 0.500 m is approximately -2.25 m/s^2.